Subgroups — C₇×#11440 #13188656

Order 1792.

450 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₇×#2112	C₂	1	2	yes	maximal
3	C₇×#1322	C₂	1	4	yes	maximal, complemented
4	C₂³×C₇×C₈	C₄	1	1	yes	characteristic
5	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
6	C₂²×C₄²×C₇	C₄	1	1	yes	characteristic
7	C₂×C₇×M₃₂	—	2	2	no	—
8	C₂×C₇×C₄⋊C₈	—	2	2	no	—
9	C₂×C₇×C₄⋊C₈	—	2	2	no	—
10	C₂×C₇×C₄⋊C₈	C₄	1	2	yes	—
11	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
12	C₇×C₂²⋊C₁₆	—	2	2	no	—
13	C₇×C₂²⋊C₁₆	C₂²	1	4	yes	—
14	#11440	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
15	C₂³×C₄×C₇	C₈	1	1	yes	characteristic
16	C₂³×C₄×C₇	C₈	1	1	yes	characteristic
17	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₇×C₈	—	2	1	no	—
19	C₂²×C₇×C₈	—	2	1	no	—
20	C₂²×C₇×C₈	—	4	2	no	—
21	C₂²×C₇×C₈	—	4	2	no	—
22	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
23	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₇×C₈	D₄	1	2	yes	—
25	C₂²×C₇×C₈	D₄	1	2	yes	—
26	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
27	C₂×C₄²×C₇	—	2	1	no	—
28	C₂×C₄²×C₇	—	2	1	no	—
29	C₂×C₄²×C₇	—	2	1	no	—
30	C₂×C₄²×C₇	—	2	1	no	—
31	C₂×C₄²×C₇	C₈	1	2	yes	—
32	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
33	C₇×M₃₂	—	4	4	no	—
34	C₂×C₇×C₁₆	—	2	4	no	—
35	C₇×C₄⋊C₈	—	2	4	no	—
36	C₇×C₄⋊C₈	—	2	4	no	—
37	C₂²×C₄⋊C₈	C₂×C₇	1	1	yes	characteristic
38	#2112	C₂×C₇	1	2	yes	—
39	#1322	C₂×C₇	1	4	yes	complemented
40	C₂⁴×C₇	C₂×C₈	1	1	yes	characteristic
41	C₂²×C₄×C₇	—	2	1	no	—
42	C₂²×C₄×C₇	—	2	1	no	—
43	C₂²×C₄×C₇	—	2	2	no	—
44	C₂²×C₄×C₇	—	2	2	no	—
45	C₂²×C₄×C₇	—	2	4	no	—
46	C₂²×C₄×C₇	—	2	1	no	—
47	C₂²×C₄×C₇	—	2	1	no	—
48	C₂²×C₄×C₇	—	4	2	no	—
49	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₇	C₂×C₈	1	1	yes	characteristic
52	C₂²×C₄×C₇	C₂×C₈	1	2	yes	—
53	C₂²×C₄×C₇	C₂×C₈	1	2	yes	—
54	C₂²×C₄×C₇	M₁₆	1	1	yes	characteristic
55	C₂²×C₄×C₇	M₁₆	1	1	yes	characteristic
56	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₇×C₈	—	2	4	no	—
58	C₂×C₇×C₈	—	2	2	no	—
59	C₂×C₇×C₈	—	4	1	no	—
60	C₂×C₇×C₈	—	4	2	no	—
61	C₂×C₇×C₈	—	4	1	no	—
62	C₂×C₇×C₈	—	4	2	no	—
63	C₂×C₇×C₈	—	4	2	no	—
64	C₂×C₇×C₈	—	4	2	no	—
65	C₂×C₇×C₈	C₂²⋊C₄	1	2	yes	—
66	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
67	C₄²×C₇	—	2	2	no	—
68	C₄²×C₇	—	2	2	no	—
69	C₄²×C₇	—	2	2	no	—
70	C₄²×C₇	—	2	2	no	—
71	C₇×C₁₆	—	4	4	no	—
72	C₂³×C₈	C₄×C₇	1	1	yes	characteristic
73	C₂³×C₈	C₂²×C₇	1	1	yes	characteristic
74	C₂²×C₄²	C₄×C₇	1	1	yes	characteristic
75	C₂×M₃₂	—	2	2	no	—
76	C₂×C₄⋊C₈	—	2	2	no	—
77	C₂×C₄⋊C₈	—	2	2	no	—
78	C₂×C₄⋊C₈	C₄×C₇	1	2	yes	—
79	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
80	C₂²⋊C₁₆	—	2	2	no	—
81	C₂²⋊C₁₆	C₂²×C₇	1	4	yes	—
82	C₂³×C₇	—	2	2	no	—
83	C₂³×C₇	—	2	2	no	—
84	C₂³×C₇	—	2	2	no	—
85	C₂³×C₇	C₂²⋊C₈	1	1	yes	characteristic
86	C₂³×C₇	C₂²⋊C₈	1	1	yes	characteristic
87	C₂³×C₇	C₂²×C₈	1	1	yes	characteristic
88	C₂×C₄×C₇	—	2	4	no	—
89	C₂×C₄×C₇	—	2	4	no	—
90	C₂×C₄×C₇	—	2	2	no	—
91	C₂×C₄×C₇	—	2	2	no	—
92	C₂×C₄×C₇	—	2	2	no	—
93	C₂×C₄×C₇	—	2	2	no	—
94	C₂×C₄×C₇	—	2	2	no	—
95	C₂×C₄×C₇	—	2	1	no	—
96	C₂×C₄×C₇	—	2	1	no	—
97	C₂×C₄×C₇	—	2	1	no	—
98	C₂×C₄×C₇	—	2	1	no	—
99	C₂×C₄×C₇	—	2	2	no	—
100	C₂×C₄×C₇	—	4	2	no	—
101	C₂×C₄×C₇	—	4	2	no	—
102	C₂×C₄×C₇	—	4	2	no	—
103	C₂×C₄×C₇	C₂²⋊C₈	1	1	yes	characteristic
104	C₂×C₄×C₇	C₂²⋊C₈	1	1	yes	characteristic
105	C₂×C₄×C₇	#99	1	2	yes	—
106	C₂×C₄×C₇	#101	1	2	yes	—
107	C₂×C₄×C₇	M₃₂	1	4	yes	complemented
108	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
110	C₇×C₈	—	2	2	no	—
111	C₇×C₈	—	2	2	no	—
112	C₇×C₈	—	4	2	no	—
113	C₂³×C₄	C₇×C₈	1	1	yes	characteristic
114	C₂³×C₄	C₇×C₈	1	1	yes	characteristic
115	C₂³×C₄	C₂×C₄×C₇	1	1	yes	characteristic
116	C₂²×C₈	—	2	1	no	—
117	C₂²×C₈	—	2	1	no	—
118	C₂²×C₈	—	4	2	no	—
119	C₂²×C₈	—	4	2	no	—
120	C₂²×C₈	C₂×C₄×C₇	1	2	yes	—
121	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
122	C₂²×C₈	C₇×D₄	1	2	yes	—
123	C₂²×C₈	C₇×D₄	1	2	yes	—
124	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
125	C₂×C₄²	—	2	1	no	—
126	C₂×C₄²	—	2	1	no	—
127	C₂×C₄²	—	2	1	no	—
128	C₂×C₄²	—	2	1	no	—
129	C₂×C₄²	C₇×C₈	1	2	yes	—
130	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
131	M₃₂	—	4	4	no	—
132	C₂×C₁₆	—	2	4	no	—
133	C₄⋊C₈	—	2	4	no	—
134	C₄⋊C₈	—	2	4	no	—
135	C₂²×C₇	—	2	8	no	—
136	C₂²×C₇	—	2	1	no	—
137	C₂²×C₇	—	2	1	no	—
138	C₂²×C₇	—	2	1	no	—
139	C₂²×C₇	—	2	1	no	—
140	C₂²×C₇	—	2	1	no	—
141	C₂²×C₇	—	2	1	no	—
142	C₂²×C₇	#324	1	2	yes	—
143	C₂²×C₇	#349	1	2	yes	—
144	C₂²×C₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
145	C₂²×C₇	C₂×M₃₂	1	2	yes	—
146	C₄×C₇	—	2	4	no	—
147	C₄×C₇	—	2	1	no	—
148	C₄×C₇	—	2	1	no	—
149	C₄×C₇	—	4	2	no	—
150	C₄×C₇	#324	1	2	yes	—
151	C₄×C₇	C₂×#99	1	1	yes	characteristic
152	C₄×C₇	C₂×#101	1	1	yes	characteristic
153	C₂⁴	C₂×C₇×C₈	1	1	yes	characteristic
154	C₂²×C₄	—	2	1	no	—
155	C₂²×C₄	—	2	1	no	—
156	C₂²×C₄	—	2	2	no	—
157	C₂²×C₄	—	2	2	no	—
158	C₂²×C₄	—	2	4	no	—
159	C₂²×C₄	—	2	1	no	—
160	C₂²×C₄	—	2	1	no	—
161	C₂²×C₄	—	4	2	no	—
162	C₂²×C₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
163	C₂²×C₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
164	C₂²×C₄	C₂×C₇×C₈	1	1	yes	characteristic
165	C₂²×C₄	C₂×C₇×C₈	1	2	yes	—
166	C₂²×C₄	C₂×C₇×C₈	1	2	yes	—
167	C₂²×C₄	C₇×M₁₆	1	1	yes	characteristic
168	C₂²×C₄	C₇×M₁₆	1	1	yes	characteristic
169	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
170	C₂×C₈	—	2	4	no	—
171	C₂×C₈	—	2	2	no	—
172	C₂×C₈	—	4	1	no	—
173	C₂×C₈	—	4	2	no	—
174	C₂×C₈	—	4	1	no	—
175	C₂×C₈	—	4	2	no	—
176	C₂×C₈	—	4	2	no	—
177	C₂×C₈	—	4	2	no	—
178	C₂×C₈	C₇×C₂²⋊C₄	1	2	yes	—
179	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
180	C₄²	—	2	2	no	—
181	C₄²	—	2	2	no	—
182	C₄²	—	2	2	no	—
183	C₄²	—	2	2	no	—
184	C₁₆	—	4	4	no	—
185	C₂×C₇	—	2	2	no	—
186	C₂×C₇	—	2	2	no	—
187	C₂×C₇	—	2	2	no	—
188	C₂×C₇	C₂×#324	1	1	yes	characteristic
189	C₂×C₇	#2112	1	1	yes	characteristic
190	C₂×C₇	C₂×#349	1	1	yes	characteristic
191	C₂³	—	2	2	no	—
192	C₂³	—	2	2	no	—
193	C₂³	—	2	2	no	—
194	C₂³	C₇×C₂²⋊C₈	1	1	yes	characteristic
195	C₂³	C₇×C₂²⋊C₈	1	1	yes	characteristic
196	C₂³	C₂²×C₇×C₈	1	1	yes	characteristic
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	2	4	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	4	2	no	—
210	C₂×C₄	—	4	2	no	—
211	C₂×C₄	—	4	2	no	—
212	C₂×C₄	C₇×C₂²⋊C₈	1	1	yes	characteristic
213	C₂×C₄	C₇×C₂²⋊C₈	1	1	yes	characteristic
214	C₂×C₄	C₇×#99	1	2	yes	—
215	C₂×C₄	C₇×#101	1	2	yes	—
216	C₂×C₄	C₇×M₃₂	1	4	yes	complemented
217	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
218	C₂×C₄	C₂×C₇×M₁₆	1	1	yes	characteristic
219	C₈	—	2	2	no	—
220	C₈	—	2	2	no	—
221	C₈	—	4	2	no	—
222	C₇	#11440	1	1	yes	characteristic, direct product factor, Sylow
223	C₂²	—	2	8	no	—
224	C₂²	—	2	1	no	—
225	C₂²	—	2	1	no	—
226	C₂²	—	2	1	no	—
227	C₂²	—	2	1	no	—
228	C₂²	—	2	1	no	—
229	C₂²	—	2	1	no	—
230	C₂²	C₇×#324	1	2	yes	—
231	C₂²	C₇×#349	1	2	yes	—
232	C₂²	C₂×C₇×C₂²⋊C₈	1	1	yes	characteristic
233	C₂²	C₂×C₇×M₃₂	1	2	yes	—
234	C₄	—	2	1	no	—
235	C₄	—	2	1	no	—
236	C₄	—	2	4	no	—
237	C₄	—	4	2	no	—
238	C₄	C₇×#324	1	2	yes	—
239	C₄	C₂×C₇×#99	1	1	yes	characteristic
240	C₄	C₂×C₇×#101	1	1	yes	characteristic
241	C₂	—	2	2	no	—
242	C₂	—	2	2	no	—
243	C₂	—	2	2	no	—
244	C₂	C₂×C₇×#324	1	1	yes	characteristic
245	C₂	C₇×#2112	1	1	yes	characteristic
246	C₂	C₂×C₇×#349	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₇	C₂×C₂²⋊C₄
commutator	C₂³	C₂²×C₇×C₈
Frattini, cosocle	C₂²×C₈	C₂³×C₇
socle	C₂²×C₇	C₂×C₂²⋊C₈
prime radical	C₂⁴×C₇	C₂×C₈
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂³×C₇×C₈ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. C₇×#1322	C₂	semidirect
14. #11440	C₇	direct (N × H)
39. #1322	C₂×C₇	semidirect
107. C₂×C₄×C₇	M₃₂	semidirect
216. C₂×C₄	C₇×M₃₂	semidirect
222. C₇	#11440	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₇×C₄⋊C₈	2
C₇×#2112	2
C₇×#1322	2
#11440	7

Subgroups — C7×#11440 #13188656

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#11440 #13188656